Curve Modifier doesn't work at all in some curves Not even deformation of shapes
I think curve called Problem Curve does really has problems. I couldn't reproduce this behaviour which is the fact that neither Fit Curve option of Array modifier doesn't work nor mesh can't be curved with it. Re-adding modifiers also won't change that.



The possible solutions are:


model another curve and use it instead of this one;
refresh several settings of Array modifier and clean existing curve.


If using the second solution, the steps are:


With Problem Curve_Model selected in settings of Array modifier set Fit Length, set some random value, then return option to Fit Curve.
In order to finish solution you should clean the curve; it has too many vertices which is overkill in this case (and for some reason it allows modifiers to work further on mesh object).
Select curve, in Edit mode select all, make any vertex active and in 3D View header choose Select > Checker Deselect operator. When vertices are selected one by one, delete them via X > Vertices. The mesh will become curved. Make sure to recalculate normals on the mesh (Ctrl+N in Edit mode with all selected).